International Journal of Solids and Structures 51 (2014) 4327–4335

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International Journal of Solids and Structures

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Characteristics of indentation cracking using cohesive zone finiteelement techniques for pyramidal indenters

http://dx.doi.org/10.1016/j.ijsolstr.2014.08.0190020-7683/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +82 42 868 4525; fax: +82 42 868 8622.E-mail address: jinhaeng@kaeri.re.kr (J.H. Lee).

Hong Chul Hyun a, Felix Rickhey a, Jin Haeng Lee b,⇑, Jun Hee Hahn c, Hyungyil Lee a

a Dept. of Mechanical Engineering, Sogang University, Seoul 121-742, Republic of Koreab Research Reactor Mechanical Structure Design Division, Korea Atomic Energy Research Institute, Daejeon 305-353, Republic of Koreac Division of Industrial Metrology, Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 January 2014Received in revised form 6 August 2014Available online 6 September 2014

Keywords:Indentation cracking testCohesive zone modelIndenter geometryFEANanoindentation

During indentation of brittle materials, cracks may be generated around the impression, depending onload conditions, material and indenter geometry. We investigate the effect of indenter geometry(centerline-to-face angle and number of edges) on crack characteristics by indentation cracking testand finite element analysis (FEA). Considering conditions for crack initiation and propagation, an FEmodel is employed featuring cohesive interfaces in zones of potential crack formation. After verificationof the FE model through comparison with experimental results for Vickers and three-sided pyramidalindentation, we study the crack morphology for diverse indenter geometries and establish a relationbetween the crack length c and the number of indenter edges nc . Together with a relation betweenindenter angle w and crack length c, we can predict the length of the crack induced by other types ofindenter from the crack size obtained with a reference indenter.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Unlike the conventional standard testing methods for assessingfracture toughness of brittle materials, indentation test can bereadily applied to small samples. In addition, the complicatedfabrication of specimens and the initial crack generation are notrequired whereas they are inevitable in the conventional methods.The finding by Palmqvist in the 1960s that load and length of thecrack that forms at the corner of the impression are directly relatedto each other, and its further development by Lawn, Evans andMarshall (LEM) in the late 1970s and early 1980s paved the roadto an readily applicable and inexpensive but contentious methodto determine Kc .

The LEM model (Lawn et al., 1980) based on Hill’s internal cav-ity analysis (Hill, 1950) requires the maximum indentation loadPmax, the hardness H, the Young’s modulus E and the crack lengthc (Fig. 1) to evaluate the fracture toughness as follows:

Kc ¼ aEH

� �1=2 Pc3=2

� �ð1Þ

where, as Anstis et al. (1981) found from Vickers indentationon some amorphous and mono-/poly-crystalline materials, the

coefficient a amounts to 0.016 ± 0.004, provided the crack is suffi-ciently long ðcla P 2Þ. The main strength of Eq. (1) is the parametersand the material properties needed to evaluate the fracture tough-ness can be directly obtained from indentation test itself. Hence,this model has been often used to evaluate the fracture toughnessof brittle materials (Pharr, 1998; Field et al., 2003; Pang et al.,2013; Zhang et al., 2013; Abdoli et al., 2014). The LEM model, how-ever, has inherent limitations and errors because complicated andcombined contact and fracture mechanics problems which defi-nitely depend on material properties are simplified and idealizedto deduce Eq. (1), and a is defined just as a function of the shapeof indenter irrelevant to material properties. In addition, the cracklength that can be measured from the surface crack provides onlylittle information on the quite complex subsurficial crack morphol-ogies including crack initiation and propagation process. The finalcrack shape may be, for instance, a shallow radial crack or a well-developed half penny-shaped crack as the result of a merging ofmedian and radial cracks, depending on material properties andindenter shape (Lee et al., 2012). Many other studies (Lankford,1982; Niihara, 1983; Laugier, 1985; Tanaka, 1987) to predict thefracture toughness of brittle materials by using similar parameters,such as the crack size a or l (Fig. 1), the maximum indentation load,the hardness, and indentation modulus, have similar problems.Hence, the previous indentation-based methods have the drawbackof being relatively inaccurate, as shown by Ponton and Rawlings

(a) (b) (c)

Fig. 1. Schematic figure of crack pattern induced by (a) four-sided and (b) three-sided pyramidal indenters; (c) definition of centerline-to-face angle w.

σmax

δmax δc

( )2 2

max1

2c c

K

E

ν σ δΓ−

= =

damage initiation

crack initiation

Fig. 2. Traction-separation law for crack simulation (Lee et al., 2012).

4328 H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335

(1989a,b), who subjected nineteen of the indentation toughnessequations including the LEM method to close scrutiny andconcluded that the evaluated fracture toughness values can varyby 20–30% and more from reference values, and the accuracyseverely depends on the material characteristics. Quin and Bradt(2007) confirmed these discrepancies and outlined the limitationsof Vickers indentation fracture toughness tests. An accurate fracturetoughness evaluation method therefore requires understanding ofthe crack evolution and its influence on indentation crackingformulas.

Numerically, crack initiation and propagation can be modeledefficiently using the Cohesive Zone Model (CZM) where a cohesiveinterface consisting of cohesive elements governed by a traction-separation law is placed in the surface where cracking is expected(Needleman, 1987; Camacho and Ortiz, 1996; Ortiz and Pandolfi,1999); the method requires, thus, preliminary knowledge of thecrack plane. Convergence difficulties can be evaded by attributinga low viscosity to the constitutive equation of cohesive elements(Gao and Bower, 2004). The CZM-based FE model suggested byLee et al. (2012) is applied here to analyze the influence of indentershape, i.e. indenter angle and number of edges, on the crackmorphology.

The paper is organized as follows: Section 2 briefly addressesindentation cracking theory and FE modeling. Results obtainedwith the CZM-based FE model are compared with Anstis et al.’sVickers indentation test data in Section 3. In Section 4, crack lengthdata for three-sized indentation tests on (100) Si and (100) Gespecimens then serve for further verification of the FE model.Finally, a relationship between crack length and indenter shapein terms of indentation load and crack length is then suggestedfrom FEA-based parameter study.

2. Cohesive zone model for cracking simulation

The cracking process is simulated by introducing a zero-thicknesscohesive interface in the plane of potential cracking. The behaviorof cohesive elements in this interface is governed by atraction-separation law, which consists of a linear elastic partwhich gives additional compliance of the FE model prior to damageand a softening part describing the stiffness degradation uponfurther loading (Fig. 2). That is, the cohesive traction T increaseswith separation d up to the point where the separation betweenthe two cohesive surfaces reaches a critical value ðrmaxÞ; beyondrmax the cohesive traction decreases to zero. Since FE results areinsensitive to the shape of the T-d law (Hutchinson and Evans,2000; Williams, 2002; Jin and Sun, 2005; Lee et al., 2012), weherein assume the bilinear form depicted in Fig. 2. The law is thendefined by three parameters, namely, damage-initiating stressrmax, corresponding damage-initiating displacement dmax and

failure displacement dc where the traction decreases to zero andboth sides of the interface are free to move independently from eachother. This relationship can be expressed by the following equation:

T ¼rmax

ddmax

when 0 6 d 6 dmax

rmaxdc�d

dc�dmaxwhen dmax 6 d 6 dc

(: ð2Þ

It should be noted that mode-I type cracks are only considered inthe present work because of the symmetric boundary conditionsof the radial/median/half-penny type cracks (Lawn, 1993). Theenergy dissipated as a result of the damage process (critical fractureenergy) is equal to the area under the traction separation curve andcan be calculated by

C ¼ K2c

E0 ¼12rmaxdc; E0 ¼

E : plane stressE=ð1� m2Þ : plane strain

�ð3Þ

where m is the Poisson’s ratio (Irwin, 1957). The plane strain elasticmodulus is used in the present work (Lee et al., 2012, Johanns et al.,2014). How to appropriately choose cohesive law parameters isminutely outlined in Lee et al. (2012).

Commercial finite element package, ABAQUS (2008) is used forthe numerical simulations of indentation as shown in Fig. 3. Usingthe bilinear cohesive interface model, parameter study isperformed varying indentation depth (or the maximum load) andindenter shape (angle and the number of edges). Geometric andload symmetries allow the use of a quarter model for indentationusing four- or eight-sided indenters and a 1/3 model for indentationusing three- or six-sided indenters [Fig. 3(a)–(c)]; both modelsconsist of approximately 87,000 elements and 97,000 nodes,respectively. A more detailed description of the mesh resolutionis described in Lee et al. (2012)’s work. For six-sided and

Fig. 3. (a) Overall FE mesh of the cohesive interface model and the boundary conditions of the (b) 1/4 model for four-sided or eight-sided, (c) 1/3 model for three-sided or six-sided, and (d) full model for three-sided pyramidal indenters.

H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335 4329

eight-sided indenters, more simple models such as 1/6 and 1/8models can be used to reduce the computational time, but we justuse 1/3 and 1/4 models because a couple of analysis cases are con-sidered. A full model is also generated for the three-sided indenterssince the crack development of the edge plane and the face plane isnot even. The full model [Fig. 3(d)] comprises 260,000 elementsand 270,000 nodes.

In the specimen interface where the crack can be initiated andpropagated, we put cohesive elements with zero thickness to sim-ulate crack initiation and separation as shown in Fig. 3. As cohesiveelements with zero thickness can have negative displacement andthere is no constraint between the cracked surfaces after the crackgenerated, we define contact constraint between two adjacentmaterial surfaces normal to indentation direction, which canprevent these surfaces from penetrating each other. Cohesiveelements share nodes with their neighboring parts although a finerdiscretization for the cohesive elements would give more accurateresults (ABAQUS, 2008).

We place contact surfaces at both material and indenter. Forthe simplicity of material properties of ceramics, the material isassumed to be elastic-perfectly plastic. The elastic diamondindenter (Young’s modulus EI = 1016 GPa, Poisson’s ratiomI = 0.07) is pressed into the top surface of the specimen whilethe bottom surface is fixed. Lee et al. (2012) ignored the frictioneffect between the material and the indenter because of thecomputation time and its second importance, but in the presentwork, friction between indenter and specimen is considered sinceit influences the stress field beneath the surface and thus thecrack length. The Coulomb friction model is applied with frictioncoefficient f = 0.2, a value that approximately applies to varioustypes of materials.

The effects of cohesive zone properties on the process zone sizehave been deeply investigated by Lee et al. (2012), and we followtheir criteria. The crack front can be where the thickness of cohe-sive element (or critical opening displacement of cohesive ele-ment) is dmax; dc , or any other values between them, and thesensitivity to the criterion decreases with process zone size. Weare only interested in well-developed half-penny cracks of whichthe process zone size is relatively small compared to the cracklength, so the definition of the crack length is not a critical problemfor well-developed cracks. In this study, we choose dc in thestrictest sense. To get the crack front line, we first measure theeach separation between adjacent two nodes of cohesive elementsalong the cohesive element thickness direction, and then welinearly interpolate the value to calculate the crack front whered ¼ dc . Because the crack can be closed (d becomes less than dc )after full separation (d P dc), it is needed to store the crack frontpositions in every increment, and compare them with those inthe updated feature. If d, which has been greater than dc in aprevious increment, is smaller than dc in the current increment,

the deformed (updated) position of the stored crack front shouldbe the new crack front in the current increment.

3. Comparison of Vickers indentation tests and FE analyses

For the verification of our FE model, five materials, i.e., Si3N4

(NC132), glass ceramic, Si3N4 (NC350), aluminosilicate glass, andsoda-lime glass, are applied to the indentation cracking simulation,and FE results are compared with Anstis et al. (1981)’s Vickersindentation experimental data. The material properties of themare listed in Table 1. Values for the Young’s modulus, the hardness,the fracture toughness Kc , and Pmax=c3=2 of indentation tests aretaken from Anstis et al. (1981), and values for the Poisson’s ratiom from Lee et al. (2012)’s work. The yield strengths ro are obtainedby trial and error method using FE analysis until the FE hardnessapproaches the given value. Indentation analyses are performedwith the maximum loads between 1 and 100 N.

From the FE analysis, it is confirmed that in well-developedcracks, Pmax=c3=2 is the material coefficient (Fig. 4). When Kc valuesare calculated with a = 0.016, Pmax=c3=2 values in Table 1 differ by12% from experimental values obtained by Anstis et al. (1981).Considering the errors associated with the crack length measure-ment and measured or guessed material properties, we can statethat the CZM-based model is reasonable. It can be also noted thatthe maximum load Pmax above which Pmax=c3=2 converges to a con-stant value varies with material properties: For Si3N4 (NC132) aconstant value is reached for Pmax P 80 N, while for aluminosilicateglass and soda-lime glass the load is much lower (Pmax 6 3 N). Inaddition, the c/a value beyond which Pmax=c3=2 becomes constantalso depends on material properties (Fig. 4), so the valid c/a regionof Eq. (1) is material-dependent. For aluminosilicate glass (AG),soda-lime glass (SLG) and glass ceramic (GC), Pmax=c3=2 convergesto a constant value at low ratios c=a 6 2.0 (Fig. 4). Generally, inmaterials with small E/H and Kc , well-developed cracks form evenat low c/a. Although the value of c/a where Pmax=c3=2 converges to aconstant value depends on the material properties, it is roughlyconstant where c/a > 2.5 in all of the 5 materials. The impressionand crack shape resulting from indentation on Si3N4 (NC132) withPmax = 100 N is depicted in Fig. 5. It can be found that thecracks appear on the surface of material and in the lower part ofindentation ‘‘well-developed cracks’’ (half-penny cracks) havebeen formed.

4. Comparison of indentation test and FE analysis forthree-sided pyramidal indenters

4.1. Indentation fracture toughness tests

The pyramidal indenter shape is characterized by twoparameters: the number of indenter edges nc (which is equal to

Table 1Comparison of the Pmax=c3=2 obtained from FEA and Vickers indentation tests (Anstis et al., 1981) on Si3N4 (NC132), glass ceramic (C9606), Si3N4 (NC350), aluminosilicate glass,and soda-lime glass materials.

Materials Material properties FE analysis (rmax = 0.5 GPa) Indentation testa

Kca (MPa m1=2 E/Ha mb ro

c (GPa) Pmax (N) c (lm) c=a Pmax=c3=2 (mN/lm3/2) Pmax=c3=2 (mN/lm3/2)

Si3N4 (NC132) 4.0 300/18.5 (16.2) 0.24 10.2 5 12.1 1.03 124.2 6010 21.9 1.33 97.720 38.5 1.66 83.940 65.8 2.00 75.060 89.3 2.29 70.980 109.1 2.39 68.4

100 129.2 2.49 67.9

Glass ceramic (C9606) 2.5 108/8.4 (12.9) 0.24 4.5 5 18.8 1.06 60.4 4310 32.9 1.30 53.120 55.5 1.50 48.340 91.9 1.80 45.350 107.1 1.93 45.0

Si3N4 (NC350) 2.0 170/9.6 (17.7) 0.24 4.5 3 12.8 1.87 30.9 335 16.7 1.66 28.8

10 23.2 2.23 27.120 32.9 2.54 26.3

Aluminosilicate glass 0.91 89/6.6 (13.5) 0.28 3.6 3 15.2 2.14 16.4 195 19.3 2.36 16.3

Soda-lime glass 0.74 70/5.5(12.7) 0.24 3.1 1 9.6 1.86 13.2 143 17.5 2.17 13.1

a Anstis et al. (1981).b Lee et al. (2012).c The yield strength obtained by trial and error using FE analysis in the present work.

c / a1.0 1.5 2.0 2.5 3.0

Pm

ax/c

3/2

(mN

/μm

3/2 )

0

30

60

90

120

150Anstis et al. (1981)

SLG

AG Si3N4 (NC350)

GC

Si3N4 (NC132)

Fig. 4. Pmax=c3=2 vs. c/a for 5 materials from FEA and literature values (dashed lines).[GC: Glass Ceramic, AG: Aluminosilicate Glass and SLG: Soda-Lime Glass]

4330 H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335

the number of cracks) and the centerline-to-face angle w[Fig. 1(c)].However, in contrast to indenters with an even number of edges,the crack morphology of the Berkovich indenter in the crack plane(cohesive interface) is not symmetric with respect to the loadingaxis. For the three-sided pyramidal indentation test and FEanalysis, indenter angles w = 35.3� (cube-corner), 45�, 55�, and65.3� (Berkovich) are used.

Indentation fracture toughness tests are conducted on (100) Siand (100) Ge specimens using three-sided pyramidal indenterswith different values for w and Pmax; radial crack length c andimpression diagonal 2a [Fig. 1(b)] are measured subsequently(values for c and a in Table 2 are average values). The indentationtests are carried out on a Nano Indenter-XP (Agilent Technologies).Indentation is performed in a load-controlled manner with loadsup to Pmax = 50 mN and loading rate v i = 0.5 mN/s. Loading ratesup to 5.0 mN/s are reported to not influence crack length results(Jang and Pharr, 2008).

Sharp indenters (w = 35.3�, 45�) however may cause consider-able chipping, especially in materials with relatively low fracturetoughness such as Ge as shown in Fig. 6. Therefore, indentationresults with w = 55� and w = 65.3� are only employed with themaximum loads Pmax = 50, 75, 100 mN. 9 tests are conducted foreach combination of w and Pmax, and crack lengths are measuredas shown in Table 2. The deviation of the measured average cracklength is below 5%.

Jang and Pharr (2008) showed that by their indentation testswith the materials of Si and Ge under the same load, a is shownto be constant regardless of w whereas c increases with decreasingw. For the same loads, i.e. the same projected impression areas, theindentation depth is higher for lower w. Consequently, the wedg-ing force acting on the crack surfaces is higher, and the crackbecomes longer. The SEM surface images in Fig. 7 taken afterindentation with Pmax = 50 mN, w = 55� on (100) Si and (100) Geshow exemplarily the star-shaped crack patterns on the surface;cracks have formed at all corners of the impression and propagatedalong the radial direction.

4.2. Indentation cracking analysis with three-sided indenters

For the simulation of the material behavior of cohesive ele-ments, we have to provide the values of the fracture toughnessKc (or the fracture energy C). In order to predict cracking behaviorof real materials, the Young’s modulus E and the yield strength ro

must be determined accurately. Oliver and Pharr (1992) suggestedthe next equation to obtain the Young’s modulus from the initialslope of the unloading branch of the indentation load–displace-ment ðP � htÞ curve (O–P method)

S ¼ dPdht

����ht¼hmax

¼ b2ffiffiffiffipp Er

ffiffiffiAp

1Er¼

1� m2� �

Eþ

1� m2I

� �EI

:

ð4Þ

Here, S, Er and A are initial slope of the unloading curve, theeffective Young’s modulus and the impression size (contact area)at Pmax, respectively; for Berkovich indenters, the geometric

x (μm)

0 50 100 150 200

z (μ

m)

-150

-100

-50

0

50

Deformed surface at unloaded stateCrack plane at unloaded state

Pmax= 100 mNVickers

c

a

x

Fig. 5. Crack morphology obtained by FEA after unloading for Si3N4 (NC132): top view (left); cut view (right).

H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335 4331

coefficient b is 1.034. Berkovich indentation tests are performedwith Pmax = 50 mN, and we get {H, E} = {12.0 GPa, 180 GPa} for Siand {H, E} = {10.0 GPa, 144 GPa} for Ge. However, the sensitivityof S to the range of regression and the difficulties of evaluation ofthe contact area may yield inaccurate values of H and E. The O–Pmethod has limitation to evaluate the contact area, and the evalu-ated contact area can be smaller than the actual one which makesthe evaluated Young’s modulus overestimated. The E value obtainedby the O–P method is therefore taken as a start value in the FEanalysis, and modified until the load–displacement (P � htÞ curvematches the experimental one. P � ht curves for Si (100) whenPmax = 50 mN are depicted in Fig. 8. When {E, ro} = {138 GPa,5.4 GPa} for Si and {E, ro} = {113 GPa, 4.7 GPa} for Ge in FEanalysis, we can obtain {H, E} = {12.0 GPa, 180.8 GPa} for Si and

Table 2Comparison of the crack length a and c/a ratio obtained from three-sided indentation and

w(�) Materials Kc (MPa m1=2) Nano-inden

P (mN)

55 Si (100) 0.7 501.0

Ge (100) 0.6 500.5

65.3 Si (100) 0.7 501.0

Ge (100) 0.6 5075

1000.5 50

(a)

Fig. 6. SEM images after indentation on (100) Ge f

{H, E} = {10.2 GPa, 143.4 GPa} for Ge by the O–P method, respec-tively, which are very close to the experimental ones. It should benoted that these combinations of material properties are not thebest choice because there exist other effects, such as indenter tipblunting, machine compliance, anisotropic behavior, and strain-hardening. However, considering our fundamental assumptionsand the standardized indentation testing method, these combina-tions would be sufficiently reasonable to compare the experimentalresults to FE ones.

The full FE model is employed to study the evolution of cracksinduced by three-sided indenters with indenter angles w = 55�and 65.3� as shown in Fig. 3(d). The maximum indentation loadis Pmax = 50 mN. The FE results of evaluated crack length c and c/aare listed in Table 2 together with fracture toughness values from

FEA on Si and Ge materials.

tation test FE analysis

c (lm) c/a c (lm) c/a

3.37 1.87 3.38 1.862.56 1.29

4.82 2.50 3.84 1.944.50 2.12

2.48 1.42 2.62 1.44– –

3.63 1.87 3.03 1.434.90 1.84 4.10 1.695.96 2.02 4.99 1.733.63 1.87 3.60 1.63

(b)

or (a) w = 35.3� and (b) w = 45� (Pmax = 50 mN).

(a) (b)

Fig. 7. SEM images after indentation with Pmax = 50 mN and w = 55� on (a) (100) Si and (b) (100) Ge.

ht (μm)0.0 0.1 0.2 0.3 0.4 0.5 0.6

P(m

N)

0

10

20

30

40

50

60

Nano indentationFEA

Si (100)

Fig. 8. Load-depth curves obtained by Berkovich indentation test and FEA for (100)Si [w = 65.3�].

xO

z

ht

B

2

3

45

O

A

1

Fig. 9. Schematic figure of possible cracking region induced by three-sidedpyramidal indenter. A is the edge plane region and B is the face plane region.

x (μm)0 5 10 15 20

z (μ

m)

-15

-10

-5

0

5

Deformed surface at unloaded stateCrack plane at unloaded state

hmax = 1.0 μmψ = 65.3o

Fig. 10. Deformed surface and crack shape at fully unloaded state for w = 65.3�three-sided pyramidal indentation and hmax = 1 lm.

4332 H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335

the literatures (Anstis et al., 1981; Jang and Pharr, 2008). For Siwith w = 55�, the deviation of the crack length between FE andexperimental values is about 30% when Kc is assumed to be1.0 MPa m1=2, but only 1.4% when Kc = 0.7 MPa m1=2. Forw = 65.3�, the deviation is 5% when Kc = 0.7 MPa m1=2 whereas nocracks have formed when Kc = 1.0 MPa m1=2.

Proceeding in the same fashion with Ge, the deviations ofcrack lengths for w = 55� and w = 65.3� is 20% and 16% whenKc = 0.6 MPa m1=2, respectively. When Kc = 0.5 MPa m1=2, thedeviation decreases to 6.6% and 0.8%, respectively. We maytherefore conclude that based on numerical results the fracturetoughness values of Si and Ge are approximately 0.7 MPa m1=2

and 0.5 MPa m1=2, respectively.

4.3. Crack evolution process of three-sided pyramidal indenters

Parameters such as indenter shape, indentation depth, andmaterial properties affect crack shape and size. As mentionedabove, in indentation with three-sided pyramidal indenters, thecrack morphology in the crack plane (cohesive interface) is notsymmetric with respect to the loading axis. Diverse possible crackshapes are sketched in Fig. 9 (crack types 2 to 5). Fig. 10 shows theshape of the crack in zone A (edge plane) obtained after Berkovichindentation (w = 65.3�) on an elastic-perfectly plastic material withE = 200 GPa, ro = 5 GPa, m = 0.3, rmax = 0.5 GPa, C = 0.0025 GPa lm(Kc = 0.74 MPa m1=2Þ. In zone A, a circular crack has formed afterindentation to hmax = 1.0 lm and subsequent unloading, whereasin zone B (face plane) the indenter has not induced any cracking.The change in crack opening displacement d (i.e. the distancebetween corresponding nodes of cracked elements) with indenta-tion depth ht is observed at a selected point (x ¼ �0:01 lm,z = �5.0 lm), and depicted in Fig. 11(a). Here, the critical displace-ment dc = 0.01 lm. Up to ht = 0.65 lm, d increases uniformly inboth zones, but for ht P 0.65 lm, d increases sharply only in zoneA while it decreases in zone B. The same is observed for otherindenter angles w = 55� and 75�; cracking occurs only in zoneA. (Table 3) We increase then the indentation depth to 2 lm, andthe d-ht graph corresponding to point (x, z) = (0.04,�10) lm isshown in Fig. 11(b). Up to ht = 1.1 lm, curves for A and B increasein a similar manner, but afterwards d increases further only in zoneA. For three-sided indenters, the number of cracks is always nc = 3,and, according to FEA results, the crack shape corresponds to cracktype 2 in Fig. 9. However, when the cracking force is concentrated

ht (μm)

0.00 0.25 0.50 0.75 1.00 1.25 1.50

δ (1

0-2μ m

)

0.0

1.0

2.0

3.0

4.0

5.0

δc = 0.01 μm

0.75 0.50

A

B

unloadingloading

ht (μm)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

δ (1

0-2μ m

)

0.0

2.5

5.0

7.5

10.0

δc = 0.01 μm

1.5 1.0

A

B

unloadingloading

(a) (b)

Fig. 11. Variation of crack displacement d with indentation depth for (a) hmax = 1 lm and (b) hmax = 2 lm at A and B points on the crack plane near the centerline.

Table 3Variation of c with respect to centerline-to-face angle for three-sided indenters.

E = 200 GPa, ro = 5 GPa, m = 0.3, KIC = 0.74 MPa m1=2, dmax=dc = 1/4

Cohesive zone material properties w(�) hmaxðlmÞ PmaxðmNÞ cðlmÞ

rmax = 0.5 GPa C ¼ 0:0025 GPa lm 55 1.0 122.4 6.8965.3 1.0 237.3 9.06

2.0 946.5 23.275 1.0 524.4 11.9

H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335 4333

on only one of three face planes or fracture toughness anisotropy isconsidered, the crack can be evolved in zone B (Johanns et al.,2014; Maerky et al., 1996).

nc

0 3 6 9 12 15

c /c

V

0.0

0.5

1.0

1.5

200 0.74400 1.05

E = 200 GPa, σo = 5 GPa, ν = 0.3

FEA

Reg.

Dukino and Swain, 1992

Kc (MPa m1/2)E (GPa)

Fig. 12. Variation of c with number of cracks nc . FE data are compared with Dukinoand Swain (1992)’s results.

5. Relationship between indenter shape and crack size

The crack morphology is mainly influenced by the number ofindenter edges nc , the crack length c (or ratio c/a) and the criticalcrack-initiating load and position. Ouchterlony (1976) establisheda relation between mode I stress intensity factor K and nc asfollows:

K ¼ k1ðncÞKF ; k1ðncÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nc2

1þ nc2p sin 2p

nc

sð5Þ

where k1 is the normalized stress intensity factor and KF ¼ F/ (pC)1/2

for a centrally loaded star crack, and F is the central expansionforce (crack opening force). For Vickers and Berkovich indenters,the following proportionalities exist

Kc / xBP=c3=2; Kc / xV P=c3=2 ð6Þ

where xB (Berkovich) and xV (Vickers) are indenter shape-dependent coefficients. (Dukino and Swain, 1992) They are relatedthrough

xB

xV¼ k1ð4Þ

k1ð3Þ: ð7Þ

According to Eq. (5), the ratio of k1 for Vickers to k1 for Berkovich isapproximately 1.073. The crack length ratio (Berkovich to Vickers)is then (1.073)2/3 = 1.048. (Dukino and Swain, 1992) Equivalently,the ratios of crack length using Vickers to crack length usingsix-sided (nc = 6), eight-sided (nc = 8), and twelve-sided ðnc = 12)pyramidal indenters are (0.8627)2/3 = 0.9062, (0.7620)2/3 = 0.8342,and (0.6310)2/3 = 0.7357, respectively, where their ideal contactareas (without pile-up/sink-in) are equivalent. Plotting the normal-ized crack length (i.e. normalized by the Vickers crack length cV )versus nc , we get the grey dash line from Dukino and Swain’s

approach in Fig. 12. As expected, c decreases to 0 for nc !1 so thatcracks are supposed never to occur for spherical indenters.

To investigate the change of crack length with crack shape, FEanalysis is performed with the models in Fig. 3(a)–(c). The chosenmaterial properties are E = 200 GPa, ro = 5 GPa, m = 0.3, rmax =0.5 GPa, C = 0.0025 GPa lm (or Kc = 0.74 MPa m1=2) and employindenters with w 69.8� (eight-sided), 69.4� (six-sided), 68� (Vick-ers), and 65.3� (Berkovich). The indenters established as such havethe same ideal contact area (without pile-up/sink-in) at the sameindentation depth. The length c and impression size 2a aremeasured at the unloaded state when the maximum indentationdepth is between hmax = 0.4 and 1.2 lm ðPmax � 30 � 330 mN).Since the ratio c/a has a constant value independent of dmax and(dmax=dc), we set dmax=dc ¼ 1=4 (Lee et al., 2012). When themaximum load Pmax is fixed, the radial length and depth of thecrack increase with decreasing nc [Fig. 13(a)]. On the other hand,at the same ht the indentation load increases with decreasing nc .This means the number of the indentation edges affects theindentation hardness. Hence, when comparing the crack lengthsof various types of indenters, the indentation load or displacementshould be well defined.

For the given material properties, when hmax P 0.8 lm(Pmax P 150 mN), well-developed cracks form and Pmax=c3=2 valuesbecome constant for all of the four (three-, four-, six- andeight-sided) indenters. Fig. 13(b) shows that the c/a thresholdwhere Pmax=c3=2 attains a constant value increases with nc .

We now turn to analyze the relation between c and nc under thecondition of well-developed crack formation. Applying the samemaximum load used in Vickers indentation to other equivalentindenters, we obtain cB=cV ; csix=cV ; ceight=cV = 1.10, 0.86, 0.76 wheresubscripts B, V, six, and eight stand for Berkovich, Vickers, six-sided,

Pmax (mN)0 100 200 300 400

c(μ

m)

0

2

4

6

8

10

12

Berkovich

Vickerssix-sided

E = 200 GPa, σo = 5 GPa, ν = 0.3, Kc = 0.74 MPa m1/2

eight-sided

c / a

1.5 2.0 2.5 3.0

Pm

ax/c

3/2 (m

N/μ

m3/

2 )

3

6

9

12

15

18

21E = 200 GPa, σo = 5 GPa, ν = 0.3, Kc = 0.74 MPa m1/2

Berkovich

Vickers

six-sided

eight-sided

)b()a(

Fig. 13. (a) c vs. Pmax and (b) Pmax=c3=2 vs. c/a for nc = 3, 4, 6, 8.

Table 4Crack length ratios when the number of indenter edges nc is assumed to be 3, 4, 6, 8. Two fracture toughness values are used in FE simulation.

ro = 5 GPa, rmax = 0.5 GPa, m = 0.3, dmax=dc = 1/4

Cohesive zone material properties Pmax (mN) cV ðlmÞ cB=cV cV=cV csix=cV ceight=cV

Kc = 0.741 MPa m1=2 (E = 200 GPa) 231.6 8.09 1.10 1.00 0.86 0.76

Kc = 1.048 MPa m1=2 (E = 400 Pa) 311.3 9.63 1.09 1.00 0.84 0.75

(cot ψ )4/90.0 0.4 0.8 1.2 1.6

c (μ m

)

0

5

10

15

20

25

75o

68o

55o

E = 200 GPa, σo = 5 GPa, ν = 0.3, Kc = 0.74 MPa m1/2

Fig. 14. Relation between the crack length c and the centerline-to-face angle w offour-sided indenters.

4334 H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335

and eight-sided indenters. Table 4 gives the values for two materialproperties. As plotted in Fig. 12, doubling values of E and C(E = 400 GPa,C = 1.048 MPa m1=2) hardly affect the ratios. Thecrack length ratio cB=cV = 1.10 (or 1.09 for doubled E and Kc)obtained from FE analysis is in good agreement with the experi-mental result of Dukino and Swain (1992), who evaluated the cracklengths on 6 materials (LinBO3,Ge,SF17,BK7,Si,SiC) with Berkovichand Vickers indention tests, and obtained 1.07 as an average ratioof the crack lengths. It is noticeable that as the crack number, nc , isincreased, the FE results are deviated from the analytical values ofDukino and Swain. Based on the data in Table 4, a formula isestablished expressing the relation between crack number nc andcrack length (normalized by cV ). Recalling that c! 0 for nc !1,we may write

c=cV ¼ k1e�k2nc ; ðk1; k2Þ ¼ ð1:3680;0:0778Þ; nc P 3 ð8Þ

where nc should be an integer. The corresponding regression line isshown in Fig. 12. Eq. (8) enables us to deduce the crack lengthsinduced by other equivalent indenters from the crack lengthinduced by a reference indenter when the maximum load is fixed.In addition, even though the maximum load is not the same as thatused in the reference indentation, we can reasonably deduce thecrack length because Pmax=c3=2 is almost constant for the well-developed half-penny crack.

Indenter angle is another factor that affects the indentationmorphology. For the indenters having the same edge numbers nc ,the crack length c can be related to indenter angle w based onLawn et al. (1980)’s observation that a is proportional to (cotw)2/3.The crack length c is therefore proportional to (cotw)4/9 fromEq. (1), and when c1 for an angle w1 is known, c2 for a different anglew2 can be obtained by

c2 ¼ c1cot w2

cot w1

� �4=9

: ð9Þ

Fig. 14 shows Eq. (9) is in accordance with results from four-sidedindentation cracking FE analysis. In conclusion, by means ofEqs. (8) and (9), a crack length evaluation for a certain combinationof (nc , w) can be extended to indenters with various nc and w.

For example, the crack length c of a three-sided pyramidal indenterwith centerline-face angel w at the maximum load Pmax;2 expectedfrom the Vickers indentation crack length cV at the maximum loadPmax;1 is

cðPmax;2Þ ¼ k1e�3k2cot w

cot 65:3�

� �4=9 Pmax;2

Pmax;1

� �2=3

cV ðPmax;1Þ ð10Þ

for the half-penny crack.To estimate the fracture toughness of material, we should relate

the crack length, material properties, and other measured values tofracture toughness. However, the crack length absolutely dependson the indenter angle and the number of edges, so an estimatedform for an indenter cannot directly use to other types ofindenters. For example, the LEM’s fracture toughness evaluationform for four-sided Vickers indentation cannot directly apply tothree-sided Berkovich or cube-corner indentation. However, if weknow the relationship between the indenter type and the cracklength, we can easily extend the evaluation form to other typesof indenters. For example, when we set up the fracture toughnessevaluation form as follows:

Kc ¼ jV m;ro; Eð ÞPmax=c3=2V ð11Þ

H.C. Hyun et al. / International Journal of Solids and Structures 51 (2014) 4327–4335 4335

where n, ro and E are material properties and the function jV isknown, The fracture toughness of Berkovich indentation can beexpressed by the following equation using Eq. (8):

Kc ¼ jB m;ro; Eð ÞPmax=c3=2B ¼ jV m;ro; Eð ÞPmax=c3=2

V

¼ jV m;ro; Eð Þðk1e�3k2 Þ3=2Pmax=c3=2

B k1; k2

¼ 1:3680;0:0778: ð12Þ

where jB is an unknown function. Therefore, even though we do notset up the coefficient function jB, we can estimate the fracturetoughness via Berkovich indentation by using the crack lengthestimation form.

6. Summary and conclusion

Using the cohesive zone FE model, we analyzed the change ofcrack length c with indenter shape (angle w and number of edgesnc) for indentation cracking tests on brittle materials. To simulateindentation cracking, cohesive interfaces were placed in the crackplanes where crack nucleation and propagation is expected. Thecohesive zone FE model was verified by comparison of the Vickersindentation test results of Anstis et al. (1981). Fracture toughnesstests using three-sided indenters were then conducted on (100) Siand (100) Ge specimens with different maximum loads and inden-ter angles. The proper fracture toughness values of (100) Si and(100) Ge were suggested by comparison of the crack characteristicsof indentation test and FE simulation results. We used a full FE modelto investigate the crack morphology of three-sided indentersbecause the crack shapes of them are not symmetric in the crackplane. When the symmetric conditions are assumed, the crack isnot developed on the face plane, but developed on the edge plane.We investigated then the relation between the crack length c andthe number of indenter edges nc based on FEA. We finally suggesteda integrated form Eq. (10) to evaluate the crack length of a certaincombination (nc , w) with a known value for other indenter geometrywhen the cracks can be assumed to be fully developed.

It should be noted that even though the crack length induced byan indenter can be simply estimated from the crack length inducedby other types of indenters, it does not directly mean the fractureevaluation form such as Eq. (1) can be simply modified in othertypes of indenters by using Eq. (8) or Eq. (9). This is because thehardness of Eq. (1) depends on the indenter shapes. Hence, toincrease the accuracy of fracture toughness values, it is necessaryto quantitatively investigate the relation of the hardness, the cracklength, and the maximum indentation load for the given indenta-tion geometry when the hardness is included in the fracture tough-ness evaluation form. However, when the hardness is not includedin the fracture toughness evaluation form, we can simply apply theintegrated form Eq. (10) for an indenter to other indenters with dif-ferent indenter angle and the number of indenter edges. Suggest-ing fracture toughness evaluation form based on the parametricstudy with the CZM-based FE model and applying Eq. (10) tovarious indenter types are our ongoing work.

Acknowledgement

This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) fundedby the Ministry of Science, ICT & Future Planning (No. NRF-2012R1A2A2A 01046480).

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